Power Domains

In denotational semantics and domain theory, power domains are domains of nondeterministic and concurrent computations.

The idea of power domains for functions is that a nondeterministic function may be described as a deterministic set-valued function, where the set contains all values the nondeterministic function can take for a given argument. For concurrent systems, the idea is to express the set of all possible computations.

Roughly speaking, a power domain is a domain whose elements are certain subsets of a domain. Taking this approach naively, though, often gives rise to domains that don't quite have the desired properties, and so one is led to increasingly complicated notions of power domain. There are three common variants: the Plotkin, upper, and lower power domains. One way to understand these concepts is as free models of theories of nondeterminism.

For most of this article we use the terms "domain" and of "continuous function" quite loosely, meaning respectively some kind of ordered structure and some kind of limit-preserving function. This flexibility is genuine; for example, in some concurrent systems it is natural to impose the condition that every message sent must eventually be delivered. However, the limit of a chain of approximations in which a message was not delivered, would be a completed computation in which the message was never delivered!

A modern reference to this subject is the chapter of Abramsky and Jung . Older references include those of Plotkin and Smyth .

Read more about Power Domains:  Power Domains As Free Models of Theories of Non-determinism, Explicit Descriptions of Power Domains, Connections With Topology and The Vietoris Space

Famous quotes containing the words power and/or domains:

    The obligation of subjects to the sovereign is understood to last as long, and no longer, than the power lasteth by which he is able to protect them.
    Thomas Hobbes (1588–1679)

    I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.
    Henry David Thoreau (1817–1862)